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lecture13 - Lecture notes 13 Date 18‐Nov‐2009 2‐D...

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Unformatted text preview: Lecture notes 13 Date: 18‐Nov‐2009 2‐D plane formulation in polar coordinates Reference: Ugural and Fenster, Sections 3.8 and 3.11 Change from Cartesian coordinates to Polar coordinates: Equations of Equilibrium in Polar Coordinates: Airy Stress function in Polar Coordinates Strain‐Displacement Relations can be obtained from the following figures Hooke’s Law does not change: Compatibility equation gives the Biharmonic Equation for the Airy Stress Function in polar coordinates: Example 1: We will write the Airy Stress function and the stresses in polar coordinates for a plate pulled in the x‐ direction by a stress σo. The Airy stress function that would give this stress state is Φ Then using sin The stresses can be obtained from the Airy Stress Function (they can also be obtained by using the stress transformation equations): Example 2: Use the Airy stress function to guess at a solution for a hole in a plate problem. Using the Airy stress function for a plate with no hole, we guess the solution for the plate with a hole to be of the form: Inserting the Airy stress function into the biharmonic equation in polar coordinates we obtain: which can be rewritten as The solutions are: The stresses are To find the constants we use the boundary conditions: ∞. In addition, they must assume and become zero because the stresses must be finite as the forms for the plate with no hole giving us the expressions for and At r=a, 0 giving us the relations: Finally, the stresses are: ...
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